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First, I think that we should write the problem as $$-\nabla(\varepsilon \nabla \phi) = \rho + \sigma\, .$$ Then, the weak form would be $$\int\limits_\Omega \varepsilon \nabla \phi \nabla w d\Omega - \int\limits_{\partial\Omega} \varepsilon w\nabla\phi\cdot\hat{n}d\Gamma = \int\limits_\Omega f w d\Omega\, .$$ As you mentioned, the second term vanishes for ...


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